I'm doing some exercises for an upcoming exam, and as part of a larger problem, I want to show that the given recursive sequence:
$$x_{k+1} = \left|x_k - \frac{x_k}{1-2M^2x_k^2}\right|$$
is monotonly increasing if $$|x_0| \geq \frac 1{2M}$$ and $M>1$.
I'm pretty sure that induction is the right approach, but I can't get the induction step to work. I tried messing around with the inverse triangle equation, but I couldn't get far.
Do you have any pointers on how to approach the problem?
$$\iff (Mx_{k+1})^2 = (Mx_k)^2 \left(1- \frac{1}{1-2(Mx_k)^2} \right)^2 \iff y_{k+1} = y_k\left(1- \frac{1}{1-2y_k} \right)^2 $$ where $y_k := (Mx_k)^2$ for $k \in \mathbb{N}.$
As $x_k \ge 0$ for $k \ge 1$, to prove the sequence $(x_k)_k$ is strictly increasing is equal to prove the sequence $(y_k)_k$ is strictly increasing.
We prove that $y_k \ge 1/4$ and $y_{k+1} \ge y_k $ for all $k \in \mathbb{N}$ by induction.
For $k = 0$, it's evident that $y_0 = M^2 x_0^2 \ge 1/4$.
Suppose the statement holds true for $k \ge 0$, we prove that it's true also for $k+1$:
$$y_{k+1} -\frac{1}{4} = y_k\left(1- \frac{1}{1-2y_k} \right)^2 - \frac{1}{4} = \frac{\left(y_k-\frac{1}{4}\right)(4y_k^2 +1)}{(1-2y_k)^2} \ge 0 \implies \color{red}{y_{k+1} \ge\frac{1}{4}}$$ $$y_{k+1} -y_k = y_k\left(1- \frac{1}{1-2y_k} \right)^2 - y_k= \frac{4y_k\left(y_k-\frac{1}{4}\right)}{(1-2y_k)^2} \ge 0 \implies \color{red}{y_{k+1} \ge y_k}$$
Then, we can conclude that the sequence $(x_k)_k$ is increasing.
Q.E.D